As the U.S. dollar surges to multiyear highs, currencies pegged to the greenback, from Saudi Arabia to Hong Kong, are under pressure.

The dollar hit a 13-year high against a basket of currencies in late November amid expectations of higher U.S. interest rates. The Federal Reserve’s two-day meeting kicks off on Tuesday, and countries around the world are watching closely as global markets move on expectations that U.S. rates will...